On the "second" Kahn--Kalai Conjecture
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918127458058240 |
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| author | Dubroff, Quentin Kahn, Jeff Park, Jinyoung |
| author_facet | Dubroff, Quentin Kahn, Jeff Park, Jinyoung |
| contents | We make progress on a conjecture of Kahn and Kalai, the original (stronger but less general) version of what became known as the ``Kahn-Kalai Conjecture" (KKC; now a theorem of Park and Pham). This ``second" KKC concerns the threshold, $p_c(H)$, for $G_{n,p}$ to contain a copy of a given graph $H$, predicting $p_c(H) = O(p_{\mathbb E}(H)\log n)$, where $p_{\mathbb E}$ is an easy lower bound on $p_c$. What we actually show is $p_{\mathbb E}^*(H)=O(p_{\mathbb E}(H)\log ^2n)$, where $p_{\mathbb E}^*$, the fractional expectation threshold, is a larger lower bound suggested by Talagrand. When combined with Talagrand's fractional relaxation of the KKC (now a theorem of Frankston, Kahn, Narayanan and Park), this gives $p_c(H)=O(p_{\mathbb E}(H)\log^3 n)$. (The second KKC would follow similarly if one could remove the log factors from the above bound on $p_{\mathbb E}^*$.) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the "second" Kahn--Kalai Conjecture Dubroff, Quentin Kahn, Jeff Park, Jinyoung Combinatorics Probability We make progress on a conjecture of Kahn and Kalai, the original (stronger but less general) version of what became known as the ``Kahn-Kalai Conjecture" (KKC; now a theorem of Park and Pham). This ``second" KKC concerns the threshold, $p_c(H)$, for $G_{n,p}$ to contain a copy of a given graph $H$, predicting $p_c(H) = O(p_{\mathbb E}(H)\log n)$, where $p_{\mathbb E}$ is an easy lower bound on $p_c$. What we actually show is $p_{\mathbb E}^*(H)=O(p_{\mathbb E}(H)\log ^2n)$, where $p_{\mathbb E}^*$, the fractional expectation threshold, is a larger lower bound suggested by Talagrand. When combined with Talagrand's fractional relaxation of the KKC (now a theorem of Frankston, Kahn, Narayanan and Park), this gives $p_c(H)=O(p_{\mathbb E}(H)\log^3 n)$. (The second KKC would follow similarly if one could remove the log factors from the above bound on $p_{\mathbb E}^*$.) |
| title | On the "second" Kahn--Kalai Conjecture |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2508.14269 |